In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, , expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function Vn(k) is the product of three terms, each depending on one coordinate alone. The ρ-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics).
Definition
Each function of this basis consists of the product of three functions:
Since all surfaces with constant ρ, φ and z are conicoid, Laplace's equation is separable in cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation can be expressed as:
The Z part of the equation is a function of z alone, and must therefore be equal to a constant:
If k is imaginary:
It can be seen that the Z(k,z) functions are the kernels of the Fourier transform or Laplace transform of the Z(z) function and so k may be a discrete variable for periodic boundary conditions, or it may be a continuous variable for non-periodic boundary conditions.
Substituting for , Laplace's equation may now be written:
Multiplying by , we may now separate the P and Φ functions and introduce another constant (n) to obtain:
Since is periodic, we may take n to be a non-negative integer and accordingly, the the constants are subscripted. Real solutions for are
The differential equation for is a form of Bessel's equation.
If k is zero, but n is not, the solutions are:
If both k and n are zero, the solutions are:
If k is a real number we may write a real solution as:
If k is an imaginary number, we may write a real solution as:
The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions:
If is the sequence of the positive zeros of then:[2]
In solving problems, the space may be divided into any number of pieces, as long as the values of the potential and its derivative match across a boundary which contains no sources.
Example: Point source inside a conducting cylindrical tube
As an example, consider the problem of determining the potential of a unit source located at inside a conducting cylindrical tube (e.g. an empty tin can) which is bounded above and below by the planes and and on the sides by the cylinder .[3] (In MKS units, we will assume ). Since the potential is bounded by the planes on the z axis, the Z(k,z) function can be taken to be periodic. Since the potential must be zero at the origin, we take the function to be the ordinary Bessel function , and it must be chosen so that one of its zeroes lands on the bounding cylinder. For the measurement point below the source point on the z axis, the potential will be:
Above the source point:
It is clear that when or , the above function is zero. It can also be easily shown that the two functions match in value and in the value of their first derivatives at .
Point source inside cylinder
Removing the plane ends (i.e. taking the limit as L approaches infinity) gives the field of the point source inside a conducting cylinder:
Point source in open space
As the radius of the cylinder (a) approaches infinity, the sum over the zeroes of Jn(z) becomes an integral, and we have the field of a point source in infinite space:
Point source in open space at origin
Finally, when the point source is at the origin,
See also
Notes
- ^ Smythe 1968, p. 185.
- ^ Guillopé 2010.
- ^ Configuration and variables as in Smythe 1968
References
- Smythe, William R. (1968). Static and Dynamic Electricity (3rd ed.). McGraw-Hill.
- Guillopé, Laurent (2010). "Espaces de Hilbert et fonctions spéciales" (PDF) (in French).